Fred, Norman, and Dave own a total of 128 comic books. If Dave owns 44 of them, what is the average (arithmetic mean) number of comic books owned by Fred and Norman?
- A. 42
- B. 44
- C. 46
- D. 48
Correct Answer & Rationale
Correct Answer: A
To find the average number of comic books owned by Fred and Norman, first determine how many comic books they collectively own. Since Dave has 44 comic books, subtract this from the total: 128 - 44 = 84. Fred and Norman together own 84 comic books. To find the average for the two, divide this number by 2: 84 ÷ 2 = 42. Option B (44) incorrectly assumes Fred and Norman have more than they actually do. Option C (46) miscalculates the average by not considering the correct total for Fred and Norman. Option D (48) similarly overestimates their combined ownership. Thus, the average is accurately calculated as 42.
To find the average number of comic books owned by Fred and Norman, first determine how many comic books they collectively own. Since Dave has 44 comic books, subtract this from the total: 128 - 44 = 84. Fred and Norman together own 84 comic books. To find the average for the two, divide this number by 2: 84 ÷ 2 = 42. Option B (44) incorrectly assumes Fred and Norman have more than they actually do. Option C (46) miscalculates the average by not considering the correct total for Fred and Norman. Option D (48) similarly overestimates their combined ownership. Thus, the average is accurately calculated as 42.
Other Related Questions
If the length of a rectangle is increased by 30% and the width of the same rectangle is decreased by 30%, what is the effect on the area of the rectangle?
- A. It is increased by 60%.
- B. It is unchanged.
- C. It is decreased by 15%.
- D. It is decreased by 9%.
Correct Answer & Rationale
Correct Answer: D
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
In the xy-plane above, the circle has center (0, 0) and AB is a diameter of the circle. What is the equation of the line passing through points A and B?
- A. y=-2/3 x
- B. y=2/3 x
- C. y=3/2 x
- D. y=4x
Correct Answer & Rationale
Correct Answer: B
The line passing through points A and B, which are endpoints of a diameter of the circle centered at (0, 0), must be a straight line that passes through the origin. Option B, \(y = \frac{2}{3}x\), represents a line with a positive slope, indicating that as x increases, y also increases, which is consistent with the properties of a diameter. Option A, \(y = -\frac{2}{3}x\), has a negative slope, suggesting a downward trend, which does not align with the upward direction of a diameter in the first quadrant. Option C, \(y = \frac{3}{2}x\), has a steeper slope than option B, which may not accurately represent the diameter's angle unless specified. Option D, \(y = 4x\), has an even steeper slope, making it unlikely to be the diameter unless A and B are positioned at extreme angles, which is not given in the problem.
The line passing through points A and B, which are endpoints of a diameter of the circle centered at (0, 0), must be a straight line that passes through the origin. Option B, \(y = \frac{2}{3}x\), represents a line with a positive slope, indicating that as x increases, y also increases, which is consistent with the properties of a diameter. Option A, \(y = -\frac{2}{3}x\), has a negative slope, suggesting a downward trend, which does not align with the upward direction of a diameter in the first quadrant. Option C, \(y = \frac{3}{2}x\), has a steeper slope than option B, which may not accurately represent the diameter's angle unless specified. Option D, \(y = 4x\), has an even steeper slope, making it unlikely to be the diameter unless A and B are positioned at extreme angles, which is not given in the problem.
Which of the following could be an equation of the line graphed in the xy-plane above?
- A. y=-x-3
- B. y=-x+3
- C. y=x-3
- D. y=x+3
Correct Answer & Rationale
Correct Answer: D
To determine the equation of the line, we analyze its slope and y-intercept. The line in the graph has a positive slope, indicating that as \(x\) increases, \(y\) also increases. Option D, \(y = x + 3\), has a positive slope of 1 and a y-intercept of 3, aligning with the graph's characteristics. Option A, \(y = -x - 3\), has a negative slope and would decrease as \(x\) increases, which contradicts the graph. Option B, \(y = -x + 3\), also has a negative slope, leading to a downward trend. Option C, \(y = x - 3\), has a positive slope but a y-intercept of -3, placing it below the graph. Thus, D is the only option that fits the observed line.
To determine the equation of the line, we analyze its slope and y-intercept. The line in the graph has a positive slope, indicating that as \(x\) increases, \(y\) also increases. Option D, \(y = x + 3\), has a positive slope of 1 and a y-intercept of 3, aligning with the graph's characteristics. Option A, \(y = -x - 3\), has a negative slope and would decrease as \(x\) increases, which contradicts the graph. Option B, \(y = -x + 3\), also has a negative slope, leading to a downward trend. Option C, \(y = x - 3\), has a positive slope but a y-intercept of -3, placing it below the graph. Thus, D is the only option that fits the observed line.
Each of the following is a solution to the equation x- 2y = 4 EXCEPT
- A. (-2,-3)
- B. (0,2)
- C. (4,0)
- D. (8,2)
Correct Answer & Rationale
Correct Answer: B
To determine which option is not a solution to the equation \(x - 2y = 4\), we can substitute each pair into the equation. - For A: \((-2, -3)\), substituting gives \(-2 - 2(-3) = -2 + 6 = 4\), which is correct. - For B: \((0, 2)\), substituting gives \(0 - 2(2) = 0 - 4 = -4\), which does not equal 4, making this option incorrect. - For C: \((4, 0)\), substituting gives \(4 - 2(0) = 4\), which is correct. - For D: \((8, 2)\), substituting gives \(8 - 2(2) = 8 - 4 = 4\), which is correct. Thus, option B is the only pair that does not satisfy the equation.
To determine which option is not a solution to the equation \(x - 2y = 4\), we can substitute each pair into the equation. - For A: \((-2, -3)\), substituting gives \(-2 - 2(-3) = -2 + 6 = 4\), which is correct. - For B: \((0, 2)\), substituting gives \(0 - 2(2) = 0 - 4 = -4\), which does not equal 4, making this option incorrect. - For C: \((4, 0)\), substituting gives \(4 - 2(0) = 4\), which is correct. - For D: \((8, 2)\), substituting gives \(8 - 2(2) = 8 - 4 = 4\), which is correct. Thus, option B is the only pair that does not satisfy the equation.